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We construct a one-parameter family of generalized Mathieu functions, whichare reduced quaternion-valued functions of a pair of real variables lying in anellipse, and which we call $\lambda$-reduced quaternionic Mathieu functions. Weprove that the $\lambda$-RQM functions, which are in the kernel of theMoisil-Teodorescu operator $D+\lambda$ ($D$ is the Dirac operator and$\lambda\in\mathbb{R}\setminus\{0\}$), form a complete orthogonal system in theHilbert space of square-integrable $\lambda$-metamonogenic functions withrespect to the $L^2$-norm over confocal ellipses. Further, we introduce thezero-boundary $\lambda$-RQM-functions, which are $\lambda$-RQM functions whosescalar part vanishes on the boundary of the ellipse. The limiting values of the$\lambda$-RQM functions as the eccentricity of the ellipse tends to zero areexpressed in terms of Bessel functions of the first kind and form a completeorthogonal system for $\lambda$-metamonogenic functions with respect to the$L^2$-norm on the unit disk. A connection between the $\lambda$-RQM functionsand the time-dependent solutions of the imaginary-time wave equation in theelliptical coordinate system is shown.

Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation